In this Appendix, we show that the expression in Eq. (REF ) can also be derived from the linear response theory [1]}, [2]}.
Given the probe Hamiltonian Eq. (REF ), we consider the correlation function
\(D(t,t^\prime )=-i\theta (t-t^\prime )\left\langle \left[f^\dagger (t)d(t),d^\dagger (t^\prime )f(t^\prime )\right]\right\rangle .\ \) 
